electromagnetism - Dielectric tensor vs. conductivity tensor in (cold) plasmas

I'm studying Waves in Cold plasmas right now, but I guess my question is generalizable. It's about the 4th Maxwell Equation in polarizable / conductive media:

$\nabla \times H = \frac{1}{c} \frac{\partial D}{\partial t} + \frac{4\pi}{c}J$

Now from our electrodynamics lectures we know that $D = \epsilon E$ with $\epsilon$ being the dielectric tensor, but also the crucial modellization of Ohm's law for conductive media $J = \sigma E$ is well within our memory. So my first reflex would now be to just plug in everything and obtain an equation for $E$ in Fourier-space and see where I get to. However this seems never to happen in the literature, and ppl as Stix (Waves in Plasmas, Chap1) or Jackson establish the relationship $\epsilon = 1 - \frac{4 \pi \sigma}{i \omega}$.

This is now what confuses me severly:
$D$ arises from the argumentation that if we have bound charges in a medium, they will linearly answer with a polarisation $P$ so that $D = E + 4\pi P$. The conductivity involves a (to be specified) steady-state modell of free charges in response to a electric field.

Although fundamentally different views, the bound and free modell seem to be connected by $\epsilon = 1 - \frac{4 \pi \sigma}{i \omega}$, but I'd be glad for some physical motivation. In the end (for cold plasmas at least), we always use $J = n m v$ and $v$ from the equation of motion to obtain a relationship between $J$ and $E$, thus obtaining $\sigma$. But there are also sources which do stuff with the dielectric tensor which I dont really understand (like Padmanabhan, Theoretical Astrophysics Vol I., Chap 9.5).

So concluding, my questions would now be the following:

• Naively plugging in $D = \epsilon E$ and $J = \sigma E$ into Maxwell 4 seems to be wrong, also due to the physical reasonings behind those modells. We can't have both in the equation. True/False?


• Why can such a relation as $\epsilon = 1 - \frac{4 \pi \sigma}{i \omega}$ exist between a static and a dynamic modell? Or am I mislead here and the $i\omega$ hints at a dynamic origin for $\epsilon$ too?

Thanks in advance, for anyone taking the time!